T(31,2)

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T(29,2).jpg

T(29,2)

T(8,5).jpg

T(8,5)

Contents

T(31,2).jpg See other torus knots

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Knot presentations

Planar diagram presentation X17,49,18,48 X49,19,50,18 X19,51,20,50 X51,21,52,20 X21,53,22,52 X53,23,54,22 X23,55,24,54 X55,25,56,24 X25,57,26,56 X57,27,58,26 X27,59,28,58 X59,29,60,28 X29,61,30,60 X61,31,62,30 X31,1,32,62 X1,33,2,32 X33,3,34,2 X3,35,4,34 X35,5,36,4 X5,37,6,36 X37,7,38,6 X7,39,8,38 X39,9,40,8 X9,41,10,40 X41,11,42,10 X11,43,12,42 X43,13,44,12 X13,45,14,44 X45,15,46,14 X15,47,16,46 X47,17,48,16
Gauss code -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15
Dowker-Thistlethwaite code 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Braid presentation
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Polynomial invariants

Alexander polynomial t^{15}-t^{14}+t^{13}-t^{12}+t^{11}-t^{10}+t^9-t^8+t^7-t^6+t^5-t^4+t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} - t^{-4} + t^{-5} - t^{-6} + t^{-7} - t^{-8} + t^{-9} - t^{-10} + t^{-11} - t^{-12} + t^{-13} - t^{-14} + t^{-15}
Conway polynomial z^{30}+29 z^{28}+378 z^{26}+2925 z^{24}+14950 z^{22}+53130 z^{20}+134596 z^{18}+245157 z^{16}+319770 z^{14}+293930 z^{12}+184756 z^{10}+75582 z^8+18564 z^6+2380 z^4+120 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 31, 30 }
Jones polynomial -q^{46}+q^{45}-q^{44}+q^{43}-q^{42}+q^{41}-q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}+q^{15}
HOMFLY-PT polynomial (db, data sources) z^{30}a^{-30}-30z^{28}a^{-30}-z^{28}a^{-32}+406z^{26}a^{-30}+28z^{26}a^{-32}-3276z^{24}a^{-30}-351z^{24}a^{-32}+17550z^{22}a^{-30}+2600z^{22}a^{-32}-65780z^{20}a^{-30}-12650z^{20}a^{-32}+177100z^{18}a^{-30}+42504z^{18}a^{-32}-346104z^{16}a^{-30}-100947z^{16}a^{-32}+490314z^{14}a^{-30}+170544z^{14}a^{-32}-497420z^{12}a^{-30}-203490z^{12}a^{-32}+352716z^{10}a^{-30}+167960z^{10}a^{-32}-167960z^8a^{-30}-92378z^8a^{-32}+50388z^6a^{-30}+31824z^6a^{-32}-8568z^4a^{-30}-6188z^4a^{-32}+680z^2a^{-30}+560z^2a^{-32}-16a^{-30}-15a^{-32}
Kauffman polynomial (db, data sources) z^{30}a^{-30}+z^{30}a^{-32}+z^{29}a^{-31}+z^{29}a^{-33}-30z^{28}a^{-30}-29z^{28}a^{-32}+z^{28}a^{-34}-28z^{27}a^{-31}-27z^{27}a^{-33}+z^{27}a^{-35}+406z^{26}a^{-30}+379z^{26}a^{-32}-26z^{26}a^{-34}+z^{26}a^{-36}+351z^{25}a^{-31}+325z^{25}a^{-33}-25z^{25}a^{-35}+z^{25}a^{-37}-3276z^{24}a^{-30}-2951z^{24}a^{-32}+300z^{24}a^{-34}-24z^{24}a^{-36}+z^{24}a^{-38}-2600z^{23}a^{-31}-2300z^{23}a^{-33}+276z^{23}a^{-35}-23z^{23}a^{-37}+z^{23}a^{-39}+17550z^{22}a^{-30}+15250z^{22}a^{-32}-2024z^{22}a^{-34}+253z^{22}a^{-36}-22z^{22}a^{-38}+z^{22}a^{-40}+12650z^{21}a^{-31}+10626z^{21}a^{-33}-1771z^{21}a^{-35}+231z^{21}a^{-37}-21z^{21}a^{-39}+z^{21}a^{-41}-65780z^{20}a^{-30}-55154z^{20}a^{-32}+8855z^{20}a^{-34}-1540z^{20}a^{-36}+210z^{20}a^{-38}-20z^{20}a^{-40}+z^{20}a^{-42}-42504z^{19}a^{-31}-33649z^{19}a^{-33}+7315z^{19}a^{-35}-1330z^{19}a^{-37}+190z^{19}a^{-39}-19z^{19}a^{-41}+z^{19}a^{-43}+177100z^{18}a^{-30}+143451z^{18}a^{-32}-26334z^{18}a^{-34}+5985z^{18}a^{-36}-1140z^{18}a^{-38}+171z^{18}a^{-40}-18z^{18}a^{-42}+z^{18}a^{-44}+100947z^{17}a^{-31}+74613z^{17}a^{-33}-20349z^{17}a^{-35}+4845z^{17}a^{-37}-969z^{17}a^{-39}+153z^{17}a^{-41}-17z^{17}a^{-43}+z^{17}a^{-45}-346104z^{16}a^{-30}-271491z^{16}a^{-32}+54264z^{16}a^{-34}-15504z^{16}a^{-36}+3876z^{16}a^{-38}-816z^{16}a^{-40}+136z^{16}a^{-42}-16z^{16}a^{-44}+z^{16}a^{-46}-170544z^{15}a^{-31}-116280z^{15}a^{-33}+38760z^{15}a^{-35}-11628z^{15}a^{-37}+3060z^{15}a^{-39}-680z^{15}a^{-41}+120z^{15}a^{-43}-15z^{15}a^{-45}+z^{15}a^{-47}+490314z^{14}a^{-30}+374034z^{14}a^{-32}-77520z^{14}a^{-34}+27132z^{14}a^{-36}-8568z^{14}a^{-38}+2380z^{14}a^{-40}-560z^{14}a^{-42}+105z^{14}a^{-44}-14z^{14}a^{-46}+z^{14}a^{-48}+203490z^{13}a^{-31}+125970z^{13}a^{-33}-50388z^{13}a^{-35}+18564z^{13}a^{-37}-6188z^{13}a^{-39}+1820z^{13}a^{-41}-455z^{13}a^{-43}+91z^{13}a^{-45}-13z^{13}a^{-47}+z^{13}a^{-49}-497420z^{12}a^{-30}-371450z^{12}a^{-32}+75582z^{12}a^{-34}-31824z^{12}a^{-36}+12376z^{12}a^{-38}-4368z^{12}a^{-40}+1365z^{12}a^{-42}-364z^{12}a^{-44}+78z^{12}a^{-46}-12z^{12}a^{-48}+z^{12}a^{-50}-167960z^{11}a^{-31}-92378z^{11}a^{-33}+43758z^{11}a^{-35}-19448z^{11}a^{-37}+8008z^{11}a^{-39}-3003z^{11}a^{-41}+1001z^{11}a^{-43}-286z^{11}a^{-45}+66z^{11}a^{-47}-11z^{11}a^{-49}+z^{11}a^{-51}+352716z^{10}a^{-30}+260338z^{10}a^{-32}-48620z^{10}a^{-34}+24310z^{10}a^{-36}-11440z^{10}a^{-38}+5005z^{10}a^{-40}-2002z^{10}a^{-42}+715z^{10}a^{-44}-220z^{10}a^{-46}+55z^{10}a^{-48}-10z^{10}a^{-50}+z^{10}a^{-52}+92378z^9a^{-31}+43758z^9a^{-33}-24310z^9a^{-35}+12870z^9a^{-37}-6435z^9a^{-39}+3003z^9a^{-41}-1287z^9a^{-43}+495z^9a^{-45}-165z^9a^{-47}+45z^9a^{-49}-9z^9a^{-51}+z^9a^{-53}-167960z^8a^{-30}-124202z^8a^{-32}+19448z^8a^{-34}-11440z^8a^{-36}+6435z^8a^{-38}-3432z^8a^{-40}+1716z^8a^{-42}-792z^8a^{-44}+330z^8a^{-46}-120z^8a^{-48}+36z^8a^{-50}-8z^8a^{-52}+z^8a^{-54}-31824z^7a^{-31}-12376z^7a^{-33}+8008z^7a^{-35}-5005z^7a^{-37}+3003z^7a^{-39}-1716z^7a^{-41}+924z^7a^{-43}-462z^7a^{-45}+210z^7a^{-47}-84z^7a^{-49}+28z^7a^{-51}-7z^7a^{-53}+z^7a^{-55}+50388z^6a^{-30}+38012z^6a^{-32}-4368z^6a^{-34}+3003z^6a^{-36}-2002z^6a^{-38}+1287z^6a^{-40}-792z^6a^{-42}+462z^6a^{-44}-252z^6a^{-46}+126z^6a^{-48}-56z^6a^{-50}+21z^6a^{-52}-6z^6a^{-54}+z^6a^{-56}+6188z^5a^{-31}+1820z^5a^{-33}-1365z^5a^{-35}+1001z^5a^{-37}-715z^5a^{-39}+495z^5a^{-41}-330z^5a^{-43}+210z^5a^{-45}-126z^5a^{-47}+70z^5a^{-49}-35z^5a^{-51}+15z^5a^{-53}-5z^5a^{-55}+z^5a^{-57}-8568z^4a^{-30}-6748z^4a^{-32}+455z^4a^{-34}-364z^4a^{-36}+286z^4a^{-38}-220z^4a^{-40}+165z^4a^{-42}-120z^4a^{-44}+84z^4a^{-46}-56z^4a^{-48}+35z^4a^{-50}-20z^4a^{-52}+10z^4a^{-54}-4z^4a^{-56}+z^4a^{-58}-560z^3a^{-31}-105z^3a^{-33}+91z^3a^{-35}-78z^3a^{-37}+66z^3a^{-39}-55z^3a^{-41}+45z^3a^{-43}-36z^3a^{-45}+28z^3a^{-47}-21z^3a^{-49}+15z^3a^{-51}-10z^3a^{-53}+6z^3a^{-55}-3z^3a^{-57}+z^3a^{-59}+680z^2a^{-30}+575z^2a^{-32}-14z^2a^{-34}+13z^2a^{-36}-12z^2a^{-38}+11z^2a^{-40}-10z^2a^{-42}+9z^2a^{-44}-8z^2a^{-46}+7z^2a^{-48}-6z^2a^{-50}+5z^2a^{-52}-4z^2a^{-54}+3z^2a^{-56}-2z^2a^{-58}+z^2a^{-60}+15za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-za^{-55}+za^{-57}-za^{-59}+za^{-61}-16a^{-30}-15a^{-32}
The A2 invariant Data:T(31,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(31,2)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Vassiliev invariants

V2 and V3: (120, 1240)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Data:T(31,2)/V 2,1 Data:T(31,2)/V 3,1 Data:T(31,2)/V 4,1 Data:T(31,2)/V 4,2 Data:T(31,2)/V 4,3 Data:T(31,2)/V 5,1 Data:T(31,2)/V 5,2 Data:T(31,2)/V 5,3 Data:T(31,2)/V 5,4 Data:T(31,2)/V 6,1 Data:T(31,2)/V 6,2 Data:T(31,2)/V 6,3 Data:T(31,2)/V 6,4 Data:T(31,2)/V 6,5 Data:T(31,2)/V 6,6 Data:T(31,2)/V 6,7 Data:T(31,2)/V 6,8 Data:T(31,2)/V 6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=30 is the signature of T(31,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
012345678910111213141516171819202122232425262728293031χ
93                               1-1
91                                0
89                             11 0
87                                0
85                           11   0
83                                0
81                         11     0
79                                0
77                       11       0
75                                0
73                     11         0
71                                0
69                   11           0
67                                0
65                 11             0
63                                0
61               11               0
59                                0
57             11                 0
55                                0
53           11                   0
51                                0
49         11                     0
47                                0
45       11                       0
43                                0
41     11                         0
39                                0
37   11                           0
35                                0
33  1                             1
311                               1
291                               1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=29 i=31
r=0 {\mathbb Z} {\mathbb Z}
r=1
r=2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}
r=5 {\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}
r=7 {\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z}
r=9 {\mathbb Z}_2 {\mathbb Z}
r=10 {\mathbb Z}
r=11 {\mathbb Z}_2 {\mathbb Z}
r=12 {\mathbb Z}
r=13 {\mathbb Z}_2 {\mathbb Z}
r=14 {\mathbb Z}
r=15 {\mathbb Z}_2 {\mathbb Z}
r=16 {\mathbb Z}
r=17 {\mathbb Z}_2 {\mathbb Z}
r=18 {\mathbb Z}
r=19 {\mathbb Z}_2 {\mathbb Z}
r=20 {\mathbb Z}
r=21 {\mathbb Z}_2 {\mathbb Z}
r=22 {\mathbb Z}
r=23 {\mathbb Z}_2 {\mathbb Z}
r=24 {\mathbb Z}
r=25 {\mathbb Z}_2 {\mathbb Z}
r=26 {\mathbb Z}
r=27 {\mathbb Z}_2 {\mathbb Z}
r=28 {\mathbb Z}
r=29 {\mathbb Z}_2 {\mathbb Z}
r=30 {\mathbb Z}
r=31 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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T(29,2).jpg

T(29,2)

T(8,5).jpg

T(8,5)